∆tξ = n X
j=1
T00(j) − n (C.19)
T00(j) est la période du cycle optique i. Pour une impulsion chirpée, elle s’exprime en fonction
de la phase temporelle quadratique αC :
T00(j) = 1 1 + αCj/2π (C.20) Pour αCj/2π << 1 on a T00(j) − 1 ≈ − αCj 2π , on a donc : ∆tξ = −αC 2π n(n + 1) 2 (C.21)
Le temps d’émission des impulsions CWE à partir d’une impulsion laser chirpée est ainsi :
τeξ≈ B + Cn(ξ) en 2/6τ2(ξ) − αC 2π n(n + 1) 2 (C.22)
Cette équation est validée par les comparaisons avec les résultats du modèle et des simula- tions PIC pour des impulsions laser chirpées positivement et négativement (voir Fig. 6.4). On retrouve notamment que la composante quadratique du délai provenant du chirp diminue la composante quadratique totale pour αC >0 : c’est la compensation du chirp femtoseconde.
C.4 Temps de naissance et vitesse de retour
La solution analytique du mouvement des électrons de Brunel relie aussi le temps de nais- sance τi = ti/T0et la vitesse de retour vr aux conditions de l’interaction (voir Eq. C.23 et C.24).
On trouve notamment le résultat non trivial vr ∝ a
1/3
0 : la vitesse de retour des électrons de Brunel qui se croisent à une profondeur donnée d ne croît pas linéairement avec l’amplitude du champ laser. A plus forte intensité, les électrons qui se croisent à la profondeur d sont arrachés plus tard du plasma (τi−0.22 ∝ −a
−1/3
0 ), ils sont donc moins accélérés par le champ laser.
τi = 0.22 − 0.29 3 v u u t √ 2d aλ0a0 (C.23) vr = √ 2c d λ0 !2/3 (a a0)1/3 (C.24)
Ces deux paramètres sont nécessaires pour estimer l’efficacité de l’émission CWE. Celle-ci dépend a priori de l’intégrale et de la hauteur du pic de densité électronique qui apparaît lors
du croisement des trajectoires électroniques. Le temps de naissance ti détermine l’amplitude du
champ laser au moment où l’électron considéré quitte le plasma ce qui permet de lui associer la densité Ni ∝ (∂E/∂t)ti. La vitesse de retour vr a un effet sur la hauteur du pic de densité car l’énergie cinétique des électrons s’oppose à l’énergie potentielle électrostatique et permet donc une meilleure concentration des électrons.
Annexe D
Publications
1. Generation of 4.3 fs, 1 mJ laser pulses via compression of circularly polarized pulses in a gas-filled hollow-core fiber,
X. Chen, A. Jullien, A. Malvache et al, Optics Letters (2009). 2. 1-mJ, sub-5-fs carrier-envelope phase-locked pulses,
X. Chen, L. Canova, A. Malvache et al, Applied Physics B Laser and Optics (2010). 3. Efficient Hollow Fiber Compression scheme for generating multi-mJ,
carrier-envelope-phase stable, sub-5 fs pulses (p.189), X. Chen, A. Malvache et al, Laser Physics (2011).
4. Multi-mJ pulse compression in hollow fibers using circular polarization (p.193), A. Malvache, X. Chen et al Applied Physics B : Laser and Optics (2011).
5. High-harmonic generation from plasma mirrors at kHz repetition rate (p.199), A. Borot, A. Malvache et al Optics Letters (2011).
6. Brunel-dominated proton acceleration with a few-cycle laser pulse,
M. Veltecheva, A. Borot, C. Thaury, A. Malvache et al, Physical Review Letters (2012). 7. Attosecond control of collective electron motion in plasmas (p.203),
A. Borot, A. Malvache et al, accepted in Nature Physics (2012).
8. Probing the density profile of a plasma mirror by coherent wake emission spectroscopy (p.209),
ISSN 1054660X, Laser Physics, 2011, Vol. 21, No. 1, pp. 198–201.
© Pleiades Publishing, Ltd., 2011. Original Text © Astro, Ltd., 2011.
198
71 1. INTRODUCTION
Since its invention over a decade ago [1, 2], HCF compressors have provided tremendous thrust to the field of ultrafast optics [3]. In a HCF filled with a rare gas at constant pressure, the spectrum of amplified, CEPlocked femtosecond laser pulses can be nonlin early broadened to support pulse durations approach ing the laser period. Adequate dispersive mirror tech nology can then be used to temporally compress the pulses, forming CEPcontrolled, high peakpower, fewopticalcycle waveforms that can drive extreme nonlinear interactions with matter [4]. Such wave forms are now routinely used to generate attosecond pulses in the extreme ultraviolet region via highhar monic upconversion in gases [5]. When focused down to wavelengthlimited spot sizes, fewcycle pulses could even be used to drive highly efficient attosecond pulse generation from relativistic laserplasma inter actions [6]. Reaching such a regime, however, demands that the ultrashort pulses have multimJ energy and very high spatial beam quality. Extended mode confinement of the laser beam inside the HCF ensures spatially homogeneous spectral broadening, excellent beam pointing stability and neardiffraction limited spatial beam quality, features that other non guided techniques, such as filamentation, cannot afford [7, 8]. However, problems arise when trying to scale standard HCF compressors beyond the mJ energy level: selffocusing and multiphoton ioniza tion prevent efficient coupling of energy into the fiber and excite highorder modes of the waveguide, thereby limiting the power density and spatial resolu tion attainable with the beam transmitted through the fiber. In practice, a standard HCF compressor with a
1The article is published in the original.
constant gas pressure may be seeded with multimJ pulses but the throughput of the device rapidly decreases, the output mode is degraded and spectral broadening becomes dominated by ionization (asym metric and blueshifted), making it generally quite dif ficult to achieve temporally and spatially clean few cycle output pulses with the desired energy.
Several techniques have successfully managed to overcome these limitations, such as applying pressure gradients across the HCF [9, 10], partial guiding using planar waveguides [11] and the use of circular polar ization [12]. Surprisingly easy to implement, circular polarization has the double benefit of increasing the critical power required for selffocusing and reducing the rate of ionization by almost one order of magni tude compared to linear polarization, resulting in a more stable output with higher temporal and spatial beam quality [13]. We recently demonstrated that cir cular polarization provides a simple way of seeding up to 2.1 mJ, 25 fs pulses into a standard HCF device to produce high quality 1 mJ, 4.3 fs compressed pulses [14]. At higher seed energies, the throughput of the HCF rapidly decreases even for circular polarization. Bohman and coworkers demonstrated the generation of 5 mJ, 5 fs pulses from a largescale pressure gradient HCF compressor device, seeded with chirped multi mJ pulses [10]. Here we show that the energy output of a compact HCF, seeded by circularly polarized pulses, can be increased by up to 60%, simply by linearly chirping the input pulses. With a standard, 1 mlong, 250 µm innerdiameter HCF, we routinely generate CEPstable (RMS ~ 250 mrad), 1.6 mJ, 4.8 fs pulses using only 3 mJ, 25 fs pulses as the seed. This straight forward approach allows us to almost double the few cycle pulse energy available from a standard HCF compressor seeded with chirpfree linearly polarized
FIBER OPTICS
Efficient Hollow Fiber Compression Scheme for Generating
MultimJ, CarrierEnvelope Phase Stable, Sub5 fs Pulses
1X. Chena, b, A. Malvachea, A. Riccia, c, A. Julliena, and R. LopezMartensa a Laboratoire d’Optique Appliquée, ENSTAParisTech, Ecole Polytechnique,
CNRS, 91761 Palaiseau Cédex, France
b Institut de la Lumière Extrême, CNRS, Ecole Polytechnique, ENSTA ParisTech, Institut d’Optique,
UniversitéParisSud, Palaiseau, France
c Thales Optronique SA, Laser Solutions Unit, Elancourt, France
*email: xiaowei.chen@ensta.fr
Received July 20, 2010; in final form, August 2, 2010; published online December 4, 2010
Abstract—We show that a standard hollowcore fiber (HCF) compressor device can be used to efficiently
compress multimJ energy laser pulses down to fewcycle duration, when seeded with linearly chirped, cir cularly polarized pulses. With this approach, we routinely generate carrierenvelope phase (CEP)locked, 1.6 mJ, 4.8 fs pulses using only 3 mJ, 25 fs pulses as the seed.
DOI: 10.1134/S1054660X11010063
LASER PHYSICS Vol. 21 No. 1 2011
EFFICIENT HOLLOW FIBER COMPRESSION SCHEME 199
pulses [14], without the need for more complex pres sures gradients schemes. Our setup features minimal losses inside the HCF (over 70% transmission) as well as excellent temporal and spatial output beam quality. The overall efficiency of the setup (53%) is limited only by the transmission of the optics downstream from the HCF. To the best of our knowledge, this is the highest CEPstable sub5 fs pulse energy reported so far.
2. EXPERIMENTAL SETUP
The 3 mJ, 25 fs seed pulses for the experiment are provided by our recently upgraded. 1 kHz, CEPstabi lized Ti:sapphire laser amplifier in Salle Noire at LOA [15]. After changing the beam polarization from linear (LP) to circular (CP), the seed pulses are launched into the HCF using an f = +1.5 m focusing mirror. The
HCF used here is a standard 1 m long, 250 µm core diameter HCF (Femtolasers GmbH) filled with a static pressure of Neon gas. The transmitted beam is changed from CP back to LP using a broadband quar ter wave plate (Femtolasers GmbH). The polarization purity of the transmitted beam, measured after a Glanlaser polarizer, is more than 99%, which is equivalent to that of the initial laser. The total pulse dispersion of the HCF setup, including the second broadband quarter wave plate and ~2.5 m air, is pre cisely compensated using a combination of broadband chirped mirrors (CMs) (Femtolasers GmbH and Lay ertech GmbH) together with a pair of thin fused silica wedges at Brewster angle. The compressed pulse dura tion is measured and optimized using a home made frequency resolved optical gating (FROG) device suit able for fewcycle pulse characterization [16].
3. EXPERIMENTAL RESULTS AND ANALYSIS
In our laser system, the spectral phase of the whole system can be arbitrarily adjusted online using an acoustooptic programmable dispersive filter (Daz zler, Fastlite), located inside the first amplification stage [17]. We systematically measured the pulse energy and spectrum at the output of the HCF as a function of initial chirp of the seed pulse. Figure 1a shows the measured transmission efficiency of the HCF as a function of Neon pressure for three different pure secondorder spectral phases of the seed pulse: 0 chirp (dashed blue line), +300 fs2 (dotted green
line), and +500 fs2 (black solid line). The energy
throughput increases with increasing input chirp and remains stable over a wider range of gas pressures. The intrinsic throughput of the HCF in the absence of gas is 80%. For an optimal input spectral phase of +500 fs2, the HCF throughput goes down to 70%
whereas it drops below 30% for chirpfree input pulses. Similar efficiencies have recently been reported for 5 mJ, 25 fs circularly polarized seed pulses in a static HCF [18]. The dramatically enhanced throughput benefits from the lower peakpower of the seed pulse (~ 2.4 times lower for +500 fs2 initial chirp,
~60 fs pulse duration), which reduces ionization induced losses inside the HCF and maintains good spatial and temporal pulse quality. Figure 1b shows that stretching the seed pulses results in moderate and more symmetric spectral broadening (solid black line). This is essential for obtaining good quality com pressed pulses. In comparison, extensive, blueshifted and strongly modulated broadening is observed in the case of chirpfree seed pulses (dashed blue line), which makes postcompression difficult because of the complex residual spectral phase and limited spec tral response of the optics downstream. Note that using negatively chirped seed pulses was not beneficial as it rapidly leads to narrow modulated output spectra.
2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 200 400 600 800 1000 1200 Neon pressure, mbar
Output pulse energy, mJ
1.0 0.5 0 1.0 0.5 0 1.0 0.5 0 1.0 0.5 0 500 600 700 800 900 1000 Wavelength, nm +500 fs2 0 chirp Ne 240 mbar Ne 500 mbar Ne 830 mbar Ne 1050 mbar +300 fs2 0 chirp +500 fs2 (a) (b)
Fig. 1. (a) Energy transmission efficiency as a function of Neon pressure for different initial secondorder spectral phase of the
seed pulses; (b) output spectrum at different gas pressures for chirpfree (dashed line) pulses and optimally chirped seed pulses (solid line).
Intensity
200
LASER PHYSICS Vol. 21 No. 1 2011
CHEN et al.
For an optimal input spectral phase of +500 fs2,
pulses transmitted through the HCF contain > 2 mJ energy. After precise dispersion compensation, at an optimal gas pressure of 1.18 bar, 4.8 ± 0.1 fs pulses with 1.6 mJ energy are routinely generated. The additional 20% energy losses incurred after the HCF originate from the optics downstream: broadband quarter wave plate, uncoated wedges and 10 bounces off chirped mirrors. More efficient chirped mirrors together with. ARcoated wedges should help significantly reduce these losses. Figure 2 shows the typically measured temporal profile and spectrum (measured by a Hamamatsu fiber spectrometer) (solid black line) together with the FROGretrieved spectral phase (dashed blue line) and spectrum (dotted black line). Here the optimally broadened spectrum supports ~4.3 fs transformlimited pulse duration. The residual spectral phase in the infrared part is due to poor dis persion compensation by the chirped mirrors in this wavelength region, implying that even shorter com pressed pulses could be achieved using chirped mirrors with broader bandwidth characteristics. However, the clean FROG trace (insert in Fig. 2a) indicates ade quate pulse compression quality. It is worth mention
ing here that the initial chirp of the seed pulses does not affect significantly the measured output spectral phase, as reported in [10], and optimal compression can be achieved by small adjustment of the wedges in the beam path (±10 fs2).
In our experiment, the octavespanning output pulse spectrum from the HCF can be directly used to measure and correct the slow CEP drift of the overall setup. A small fraction of the postcompressed beam is picked off by a broadband beam splitter and focused into a home made fto2f interferometer composed of a 0.5 mm thick BBO and a broadband polarizer. Inter ference fringes are acquired by a Thorlabs SPx spec trometer and analyzed using commercial software (Menlo Systems GmbH). Feeding the calculated CEP drift back to the Menlo electronics for the oscillator, the CEP of the whole laser system including the HCF compressor can be stabilized. A typical standard CEP deviation measurement is shown in Fig. 3a, indicating a RMS phase error of 250 mrad (10 shot averaging, 100 ms cycleloop time).
Finally, the generated sub5 fs laser beam shows excellent spatial quality as commanded by the mode
1.0 0.8 0.6 0.4 0.2 0 1.0 0.8 0.6 0.4 0.2 0 450 400 350 300 −40 0 40 Delay, fs Wavelength, nm −30 −20 −10 0 10 20 30 6 5 4 3 2 500 600 700 800 900 1000 τFWHM= 4.8 fs (a) (b)
Intensity, arb. units
Time, fs Wavelength, nm
Phase, rad
Fig. 2. Spectrotemporal characterization of the sub5 fs compressed pulses: (a) retrieved FROG trace (insert) and reconstructed
temporal profile; (b) FROGretrieved spectrum (dotted line), and spectral phase (dashed line), compared with the measured spectrum (solid line).
2 1 0 −1 −2 −3 3 0 100 200 300 Time, s CEP drift, rad
250 mrad RMS 6 5 4 3 2 1 0 1 2 3 4 5 6 1.0 0.8 0.6 0.4 0.2 0 Horizontal axis, µm Vertical axis, µ m Spatial intensity, arb. units
Fig. 3. (a) CEP drift of the 1.6 mJ, sub5 fs pulses after the HCF compressor; (b) measured spot size of the sub5 fs laser beam at
the focus of a f/1.7, 30° offaxis parabola.
(a) (b)
LASER PHYSICS Vol. 21 No. 1 2011
EFFICIENT HOLLOW FIBER COMPRESSION SCHEME 201
confinement of the HCF. The measured spot size (Fig. 3b) at the focus of a f/1.7, 30° offaxis parabola is under 1.8 µm (beam waist at 1/e2 peak intensity) with
more than 70% of the energy contained within the pulse spatial envelope (at 1/e2 peak intensity) and
without any prior wavefront correction. These condi tions correspond to focused intensity above 2 × 1018W/cm2, which is close to relativistic intensities for
800 nm light.
4. CONCLUSIONS
In conclusion, we present that by chirping the ini tially circularly polarized seed pulses, the energy throughput of a compact HCF compressor with con stant gas pressure can be increased by more than 60% and consequently, by up to almost 100% compared to seeding with linearly polarized, chirpfree pulses [14], with no added experimental complexity. High spatial quality, CEPstable, 1.6 mJ, sub5 fs pulses are rou tinely generated using only 3 mJ, 25 fs pulses from the laser. Better optics downstream should provide higher throughput and improved compression efficiency. We believe this approach could be successfully combined with the pressure gradient technique to achieve high quality pulse compression at the multiTW peak power level.
ACKNOWLEDGMENTS
Financial support from the CNRCCNRS 2007 program and the Agence Nationale pour la Recher che, through program ANR09JCJC0063 (UBI CUIL) is gratefully acknowledged.
REFERENCES
1. M. Nisoli, S. De Silvestri, and O. Svelto, Appl. Phys. Lett. 68, 2793 (1996).
2. S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, C. Spielmann, F. Krausz, and K. Ferencz, Opt. Lett. 22, 1562 (1997).
3. T. Brabec and F. Frausz, Rev. Mod. Phys. 72, 545 (2000).
4. A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fießs, V. Pervak, L. Veisz, V. S. Yakov lev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, New. J. Phys. 9, 242 (2007).
5. C. Vozzi, F. Calegari, F. Ferrari, M. Lucchini, S. De Sil vestri, O. Svelto, G. Sansone, S. Stagira, and M. Nisoli, Laser Phys. Lett. 6, 259 (2009).
6. J. Nees, N. Naumova, E. Power, V. Yanovsky, I. Aokolov, A. Maksimchuk, A. W. Bahk, V. Chvykov, G. Kalintchenko, B. Hou, and G. Mourou, J. Mod. Opt. 52, 305 (2005).
7. L. Gallmann, T. Pfeifer, P. M. Nagel, M. J. Abdel, D. M. Neumark, and S. R. Leone, Appl. Phys. B 86, 561 (2007).
8. A. Zaïr, A. Guandalini, F. Schapper, M. Holler, J. Bieg ert, L. Gallmann, A. Couairon, M. Franco, A. Mysyr owicz, and U. Keller, Opt. Express 15, 5394 (2007). 9. S. Bohman, A. Suda, M. Kaku, M. Nurhuda, T. Kanai,
S. Yamaguchi, and K. Midorikawa, Opt. Express 16, 10684 (2008).
10. S. Bohman, A. Suda, T. Kanai, S. Yamaguchi, and K. Midorikawa, Opt. Lett. 35, 1887 (2010).
11. S. Akturk, C. L. Arnold, B. Zhou, and A. Mysyrowicz, Opt. Lett. 34, 1462 (2009).
12. S. Ghimire, B. Shan, C. Wang, and Z. Chang, Laser Phys. 15, 838 (2005).
13. A. Malvache, X. Chen, C. Durfee, and R. LopezMar tens, manuscript in preparation.
14. X. Chen, A. Jullien, A. Malvache, L. Canova, A. Borot, A. Trisorio, C. Durfee, and R. LopezMartens, Opt. Lett. 34, 1588 (2009).
15. X. Chen, L. Canova, A. Malvache, A. Jullien, R. LopezMartens, C. Durfee, D. Papadopoulos, and F. Druon, Appl. Phys. B 99, 149 (2010).
16. S. Akturk, C. D’Amico, and A. Mysyrowicz, JOSA B
25, A63 (2008).
17. L. Canova, X. W. Chen, A. Trisorio, A. Jullien, A. Assion, G. Tempea, N. Forget, T. Oksenhendler, and R. LopezMartens, Opt. Lett. 34, 1333 (2009). 18. A. Anderson, G. Tempea, M. Hofer, T. Prikoszovitz,
Z. Cheng, T. Le, and A. Assion, Proc. Of SPIE 7578, 75781T (2010).
Appl Phys B (2011) 104:5–9 DOI 10.1007/s00340-011-4663-4
R A P I D C O M M U N I C AT I O N
Multi-mJ pulse compression in hollow fibers using circular
polarization
A. Malvache· X. Chen · C.G. Durfee · A. Jullien · R. Lopez-Martens
Received: 12 May 2011 / Revised version: 14 June 2011 / Published online: 15 July 2011 © Springer-Verlag 2011
Abstract We develop a numerical model to explore the
polarization-dependent compression of multi-mJ laser pulses in a gas-filled hollow fiber. We show how losses and instabilities due to cycling of pulse energy between fiber modes can be efficiently minimized using circularly po- larized light and adjusting simple experimental parameters such as pulse energy, chirp and gas pressure. This should help scale the peak power of few-cycle pulses available for high-field experiments using standard hollow fiber compres- sors. We also discuss the limits of this approach.
1 Introduction
Temporal compression of optical pulses in a gas-filled hollow-core fiber (HCF) [1] is a key technique for gener- ating few-cycle optical waveforms that can then be used for driving and probing attosecond electronic processes in mat- ter [2]. Beam guiding in the fundamental mode of the fiber offers exceptional compression fidelity for sub-mJ energy pulses. For multi-mJ pulses, however, nonlinear focusing and plasma formation excite higher-order fiber modes and severely degrade the transmitted beam quality. This impedes the generalization of this technique to nowadays widely available multi-mJ femtosecond laser technology. Pressure gradients can be applied across the fiber to avoid transverse
A. Malvache ()· X. Chen · A. Jullien · R. Lopez-Martens Laboratoire d’Optique Appliquée, ENSTA ParisTech, Ecole Polytechnique, CNRS, 91761 Palaiseau Cedex, France e-mail:arnaud.malvache@ensta-paristech.fr
Fax: +33-169-319996 C.G. Durfee
Physics Department, Colorado School of Mines, 1523 Illinois St., Golden, CO 80401, USA
beam nonlinearities up to quite pulse high energies [3] but this approach significantly adds to the size and complexity of the device.
Recently, we demonstrated efficient multi-mJ pulse com- pression using a standard HCF device seeded with circular polarization (CP) rather than linear polarization (LP) [4,5]. In this high-intensity regime, we consistently observe oscil- lations of plasma fluorescence intensity (Fig. 1) along the fiber triggered by variations of beam intensity inside the fiber core. For LP, the oscillations are chaotic and the com- pressed pulse fidelity is low. With CP, the intensity oscil- lations follow a simple periodicity and allow high-quality compression. The periodic cycling of the intensity results from beating between fiber modes that are nonlinearly cou- pled [6,7]. These results suggest that stronger and more localized ionization for LP drives the coupling of higher- order modes. They also show that one must carefully bal- ance the nonlinearities in order to minimize losses and in- stabilities arising from uncontrolled beating of fiber modes. In this work, we develop a detailed propagation model that describes the influence of both nonlinear index and ioniza- tion cross-mode coupling on the quality of HCF compres- sion. While the role of lower-order mode beating due to the nonlinear index has been addressed [6, 7], our model includes polarization-dependent ionization and high-order modes. Unlike in [8], we derive an explicit nonlinear equa- tion that reduces the problem into one dimension and we take into account the frequency dependence of the ioniza- tion nonlinearity. This method highlights the mismatched propagation of the modes which plays a key role. We use the model to show how, with CP, one can optimize multi-mJ